Is the Big Rip Unreachable?

Physics Letters B 785 (2018) 132–135

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Physics Letters B

www.elsevier.com/locate/physletb

Is the Big Rip unreachable?

Konstantinos Dimopoulos

Consortium for Fundamental Physics, Physics Department, Lancaster University, Lancaster LA1 4YB, UK a r t i c l e i n f o a b s t r a c t

Article history: I investigate the repercussions of particle production when the Universe is dominated by a hypothetical Received 15 July 2018 phantom substance. I show that backreaction due to particle production prevents the density from Received in revised form 2 August 2018 shooting to inﬁnity at a Big Rip, but instead forces it to stabilise at a large constant value. Afterwards Accepted 3 August 2018 there is a period of de-Sitter inﬂation. I speculate that this might lead to a cyclic Universe. Available online 24 August 2018 © 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license Editor: M. Trodden (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

A phantom substance is defined as a fluid which violates the perature the Hawking temperature of de Sitter space T = H/2π null energy condition because its pressure is p < −ρ, where ρ [1–7]1 is its density. Such a hypothetical substance is also called exotic 2 4 matter, and it is necessary for keeping a wormhole traversable. π gr H ρgr = q g∗ , (1) In cosmology, if the Universe content is dominated by phantom 30 2π density, then the latter increases and at some finite time tmax it gr becomes infinite, in a singularity called the Big Rip. This is be- where g∗ is the number of effective relativistic degrees of free- cause the Universe undergoes super-inflation where the acceler- dom which undergo particle production (i.e. are light and non- ated expansion gives rise to an event horizon, whose dimensions conformally invariant) and q ∼ 1is some constant factor due to shrink to zero at tmax, ripping apart all structures, from galaxies to the fact that the resulting ρgr is not actually thermal. atoms. Thus, the Friedmann equation obtains an additional term of the But is it so? So far, most of the studies of the dynamics of form the Universe when undergoing accelerated expansion have ignored 2 8π G 4 particle production due to the event horizon, because it is usually H = ρ + CH , (2) 3 negligible, in inflation for example. In this work, it is argued that particle production cannot be ignored when the accelerated ex- where ρ is the density of the substance which causes the acceler- pansion is driven by phantom density. Backreaction due to particle ated expansion and, in view of Eq. (1), we have production renders the Big Rip singularity unreachable. 4 8π G qG gr We work in Einstein gravity only. In the following, we use CH = ρgr ⇒ C = g∗ . (3) natural units, where c = h¯ = 1 and Newton’s gravitational con- 3 180π = −2 = × 18 stant is 8π G mP , with mP 2.43 10 GeV being the reduced Strictly speaking, the above are true for ρ, H constant, which Planck mass. For simplicity, isotropy and spatial flatness is as- results in quasi-de Sitter inflation. In this case, the CH4 contri- sumed throughout. bution in the Friedmann equation (2)is negligible and is usually The existence of an event horizon during accelerated expan- ignored (but see Ref. [9]). However, one expects the same phe- sion results in particle production of all light (meaning with mass nomenon to take place in any kind of accelerated expansion, since smaller than the Hubble rate m < H) non-conformally invariant gravitational particle production occurs whenever there is an event fields (e.g. a light scalar field, such as the inflaton). During quasi- horizon (as with black hole radiation, for example). Thus, qualita- 4 de Sitter inflation, particle production results in the gravitational tively, the same source term CH would appear in the Friedmann equation as H sets the scale of the accelerated expansion and generation of density of the order of thermal density with tem-

1 Note that, particle production is a non-equilibrium effect and in an isotropic E-mail address: [email protected]. universe its classical counterpart is bulk viscosity [8]. https://doi.org/10.1016/j.physletb.2018.08.040 0370-2693/© 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. K. Dimopoulos / Physics Letters B 785 (2018) 132–135 133 √ √ √ gr C ∝ g∗ as in Eq. (3).2 This term would become important if the 1 2 + 2CH 2 − 1 1 3 √ ln √ √ √ − 2 1 − √ cause of accelerated expansion is a phantom substance. 2 2 − 2CH 2 + 1 2CH If ρ is the density of a phantom substance then the Fried- | + | mann equation (ignoring the extra term CH4, because it can be = 3 1√ w − ˙ (tmax t), (7) negligible at first) results in ρ˙ > 0 and H > 0. This results in super- 2C inflation which leads to the Big Rip singularity when ρ →∞ in = finite time. However, the growth of H means that the CH4 in the where tmax corresponds to the time when ρ ρmax. Note that Friedmann equation (2)will eventually become important, because √Eq. (4)(with √the negative sign in the brackets) implies that = − − 1/2 ≡ it increases faster than the H2 term. This may halt the growth of 2CH (1 1 u) , where we have defined u ρ/ρmax, i.e. = ρ and prevent the Big Rip from happening [12,13]. u(tmax) 1. √ When H Hmax = 1/ 2C, the last term in the left-hand-side The solution to Eq. (2)is −1 3 of the above dominates and we find H = |1 + w|(tmax − t) 2 as with standard phantom dark energy, only the time tmax does 1 32π GC ∞ H2 = 1 ± 1 − ρ . (4) not denote the Big Rip, i.e. ρ(tmax) , but instead we have 2C 3 ρ(tmax) = ρmax which is finite and given by Eq. (5). The solution in Eq. (7)only applies for t ≤ tmax. At tmax the 2 = = For small density, the above equation results in either H 1/C solution becomes zero.6 We can investigate what happens when constant or H2 = (8π G/3)ρ, which is the usual Friedmann equa- t > tmax by considering the continuity equation (6), which gives tion. The latter solution corresponds to the negative sign in the brackets. However, the above also shows that there is a maximum 3|1 + w| u˙ = 3|1 + w|Hu ⇒ u˙ = √ > 0 , (8) possible value of ρ, which is ρmax = 3/32π GC, in √which case there max = 2C is a maximum value of the Hubble rate Hmax 1/ 2C. This means √ that the Big Rip is unreachable, prevented by the backreaction of where u˙ max ≡ u˙(tmax) and we used that Hmax = 1/ 2C and the gravitational particle production.4 u(tmax) = 1. Thus, there is a tendency for the density to be- Using Eq. (3), we can obtain an estimate of the maximum den- come larger than ρmax, which, however, is forbidden by Eq. (4). sity This is really because Eq. (2)is not valid any more. Indeed, for ρ > ρmax, the density of the gravitationally produced particles 2 would be larger than the phantom density itself, which cannot = 3 = 135 = 3Hmax ρmax gr happen (where is the energy coming from?). 32π GC 8qG2 g∗ 16π G This suggests that the continuity equation needs augmenting, √ 1/4 ⇒ 1/4 = 30 since the removal of energy by gravitational particle production is ρmax 6π gr mP . (5) qg∗ not considered in Eq. (8); it is implicitly assumed negligible. To take this into account we introduce a negative source term on the 1/4 gr Thus, ρmax can be smaller than the Planck scale only when g∗ is right-hand-side of Eq. (6), which becomes (see also Ref. [15]) 1/4 very large. Considering string theory, we can reduce ρmax to the gr 9|1 + w|C string scale ∼ 1017 GeV by considering g∗ ∼ 105 or so. We assume 5 ρ˙ − 3|1 + w|Hρ =−3|1 + w|Hρgr =− H . (9) 1/4 8π G that ρmax < mP such that quantum gravity considerations can be 5 ignored. 4 The form of this term is given by δρ/δt, where δρ =−ρgr ∝ H − Exactly how the density evolves can be revealed by the study as in Eq. (1)and δt ∼ H 1 is the Hubble time. This is because the of the continuity equation relativistic particles produced gravitationally in a Hubble time are diluted by the accelerated expansion and replenished by the parti- ρ˙ + 3(1 + w)Hρ = 0 , (6) cles produced in the following Hubble time, meaning that δρ ∝ H4 − is produced gravitationally per Hubble time δt ∼ H 1. The precise −1 where w = p/ρ < −1is the barotropic parameter of the phantom value is δt = H /3|1 + w|, which results in u˙ max = 0. substance. For simplicity, we consider w = constant. Using Eq. (4) Combining Eqs. (4) and (9)we may write the continuity equa- with the negative sign in the brackets, Eq. (6)can be analytically tion as solved to give √ √ 3/2 u˙ = 6|1 + w|Hmax 1 − u(1 − 1 − u) , (10) √ 2 = 1 | + = ˙ = In general, in accelerated expansion the Hawking temperature is T 2 3w where Hmax 1/ 2C. From the above it is evident that umax 0 1| H 2 [10]. The numerical factor 1 |3w +1| can be incorporated into q in Eq. (1). ( / π) 2 as expected. Also note that, in the limit u 1the above becomes 3 In principle, other terms involving the derivatives of the Hubble rate, such as ¨ 2 ˙ ˙ 2 4 √ H H, H H or (H) may also be important, additionally to the H term considered 3 3/2 ˙ 2 ˙ | + | | + | here. Such terms are negligible in quasi-de Sitter inflation because |H| H , but u 2 1 w 2Hmaxu 3 1 w Hu, (11) this would not necessarily be so for a phantom substance. However, in Ref. [11]it is shown that the density of such derivative terms is proportional to α, where the which is the usual continuity√ equation (cf. Eq. (8))√ and we consid- 2 L = 1 2 + 2 = − − 1/2 1/2 latter is the coefficient of R in a modified gravity Lagrangian 2 mP R αR . ered that H Hmax(1 1 u) Hmaxu / 2in this limit. = In this work we consider Einstein gravity only, which means α 0and these terms Note that in the limit u 1we recover the Friedmann equation√ as are absent. expected because H2 1 H2 u = 8π Gρ/3, where H = 1/ 2C 4 A similar result was obtained in Ref. [14], where the effect of the conformal 2 max max and ρ is given by Eq. (5). anomaly was taken into account, which can also produce a contribution δρ ∝ H4. max 5 In a quantum gravity setup, the H4 correction in the Friedmann equation (2) might be the leading order to terms proportional to H6 or H8 for example. 4 6 4 Such terms would grow even faster than the H . However, we expect such terms In view of Eqs. (1)and (5), we have ρgr ≤ ρ ⇔ 8π Gρ/3 ≥ CH . This means to be Planck suppressed such that they always remain subdominant because that we cannot√ connect√ with the positive branch of H2 in Eq. (4), which ranges 4+n n 4 ≥ ≤ 1/4 4 H /mP < H for all n 1since H Hmax mP because ρmax < mP . between 1/ 2C and 1/ C, because this requires 8π G/3ρ < CH . 134 K. Dimopoulos / Physics Letters B 785 (2018) 132–135

Fig. 2. Schematic evolution of the phantom density ρ in a hypothetical cyclic scenario. ρ increases with time approximating a constant ρmax, which once assumed leads to a period of inﬂation. Then, some unknown process reduces drastically the phantom density such that it becomes dark energy, until another cycle begins.

stantially altered. In many cyclic models, for example the Mixmas- ter universe in Ref. [21], the cumulative anisotropic effects grow

Fig. 1. Evolution of u = ρ/ρmax as time approaches tmax. The Big Rip is avoided and and become overwhelming, possibly destabilising the cyclic be- the density approaches a constant value ρmax. haviour. In contrast to the Mixmaster scenario however (and other similar cyclic models, e.g. Ref. [22]), the cyclic universe considered The solution to Eq. (10)is here does not involve a contracting phase (which would enlarge the anisotropy). The total density of the Universe is growing and ρ −2 2 falling but the expansion never halts or reverses itself. Thus, in this = u = 1 − 1 − 3 |1 + w|H (t − t) + 1 . (12) ρ 2 max max case, any existing anisotropy might remain negligible, although this max needs to be investigated. If we consider the limit t tmax and the condition Hmaxtmax 1 We have discussed the repercussions of particle production we find that the above asymptotes to the value ρ → 1/6(1 + when the Universe is dominated by a hypothetical phantom sub- 2 2 w) tmax. This is the same value one obtains in the limit t stance. We have argued that backreaction due to particle produc- tmax for standard phantom dark energy, for which ρ = 1/6(1 + tion prevents the density from shooting to infinity at a Big Rip, but 2 2 w) (tmax − t) . instead forces it to stabilise at a large constant value, which could The evolution of the density ρ until the time tmax is shown be near the string scale or the scale of grand unification, which is in Fig. 1. As we approach tmax, instead of shooting to infinity, the a little lower. After assuming its constant maximum value there is backreaction due to the particle production forces the growth of ρ a period of de-Sitter inflation. This might lead to a cyclic Universe, to be halted and the latter gently reaches ρmax at t = tmax. What provided some unknown mechanism terminates inflation and dras- happens afterwards? Well, the√ constant values ρ = ρmax (given by tically reduces the phantom density, giving rise to the thermal bath Eq. (5)) and H = Hmax = 1/ 2C are solutions to Eqs. (2) and (10) of the hot big bang. so that the density and the Hubble parameter remain constant. We have considered Einstein gravity and ignored quantum Thus, the Universe undergoes de-Sitter inflation, even though it is gravity corrections taking the maximum density to remain sub- filled at equal parts by a phantom substance with w < −1 and ra- Planckian.10 We have not introduced a specific model of phantom diation due to particle production which is constantly diluted and dark energy,11 in order to emphasise that our treatment and re- ∼ 4 replenished so that it has a constant net density ρgr Hmax. sults are generic and due to the existence of an event horizon We can also envisage an interesting hypothetical scenario, in accelerated expansion, which leads to particle production as in where, due to an unknown process, the phantom substance black holes. In that sense, our finding that the Big Rip is unreach- drastically decays into a minute residual density with value able, seems unavoidable. − (10 3 eV)4. Its decay products reheat the Universe and the hot big bang ensues. Eventually the phantom density takes over once Acknowledgements more, originally as dark energy,7 but its density soon increases rapidly up to the scale of grand unification (1016 GeV)4 or so, I would like to thank P. Anderson and G. Rigopoulos for dis- such that in gives rise to another boot of de-Sitter inflation.8 We cussions. My research is supported (in part) by the Lancaster– thus might have a cyclic Universe, which is schematically shown Manchester–Sheffield Consortium for Fundamental Physics under in Fig. 2.9 If the scenario can be embedded in string theory [20], STFC grant: ST/L000520/1. then one might wonder if decompactification can happen when the density grows comparable to the string scale. Afterwards the References extra dimensions might compactify at a different Calabi–Yau, im- plying that the laws of physics might be different in every cycle. [1] L. Parker, Phys. Rev. 183 (1969) 1057. This might lead to a multiverse in time and not in space. [2] R.H. Brandenberger, R. 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